The differential equation of the curve for which the point of tangency (closer to the x - axis) divides the segment of the tangent between the coordinate axes in the ratio `1:2`, is

Options

A. `xdy=2ydx`
B. `xdy=ydx`
C. `xdy+2ydx=0`
D. `xdy+ydx=0`

Answer

To solve the problem, we need to find the differential equation of the curve for which the point of tangency divides the segment of the tangent between the coordinate axes in the ratio 1:2.

### Step-by-Step Solution:

1. **Understanding the Tangent Line**:
The equation of the tangent line at a point \((x_0, y_0)\) on the curve can be written as:
\[ y - y_0 = m(x - x_0) \]
where \(m\) is the slope of the tangent at that point.

2. **Finding the Intercepts**:
- **X-intercept (A)**: Set \(y = 0\) in the tangent equation:
\[ 0 - y_0 = m(x - x_0) \implies x_A = x_0 - \frac{y_0}{m} \]
- **Y-intercept (B)**: Set \(x = 0\) in the tangent equation:
\[ y - y_0 = m(0 - x_0) \implies y_B = y_0 + mx_0 \]

3. **Coordinates of A and B**:
- The coordinates of point A (X-intercept) are \(\left(x_0 - \frac{y_0}{m}, 0\right)\).
- The coordinates of point B (Y-intercept) are \((0, y_0 + mx_0)\).

4. **Using the Section Formula**:
The point \(P\) that divides the segment \(AB\) in the ratio \(1:2\) can be calculated using the section formula:
\[ P = \left(\frac{2x_A + 1 \cdot 0}{1 + 2}, \frac{2 \cdot 0 + 1(y_B)}{1 + 2}\right) \]
This gives:
\[ P = \left(\frac{2\left(x_0 - \frac{y_0}{m}\right)}{3}, \frac{y_0 + mx_0}{3}\right) \]

5. **Setting up the Ratio**:
Since \(P\) divides \(AB\) in the ratio \(1:2\), we can write:
\[ y_P = \frac{y_B}{3} = \frac{y_0 + mx_0}{3} \]
and
\[ y_P = \frac{2 \cdot 0 + 1(y_B)}{3} = \frac{y_0 + mx_0}{3} \]

6. **Relating Slope and Coordinates**:
From the geometry of the problem, we know that the slope \(m\) can be expressed as:
\[ m = \frac{dy}{dx} \]
Thus, we have:
\[ 3y = y_0 + mx_0 \]

7. **Substituting for m**:
Rearranging gives:
\[ 3y = y_0 + \frac{dy}{dx}x_0 \]
From the earlier steps, we can express \(m\) in terms of \(x\) and \(y\):
\[ m = -\frac{2y}{x} \]

8. **Final Differential Equation**:
Substituting \(m\) into the equation gives:
\[ x \frac{dy}{dx} + 2y = 0 \]
This can be rearranged to form the final differential equation:
\[ x \frac{dy}{dx} + 2y = 0 \]

### Conclusion:
The differential equation of the curve is:
\[ x \frac{dy}{dx} + 2y = 0 \]